# transformations replaced by symmetry

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## 8 matching pages

##### 1: 19.15 Advantages of Symmetry
Symmetry in $x,y,z$ of $R_{F}\left(x,y,z\right)$, $R_{G}\left(x,y,z\right)$, and $R_{J}\left(x,y,z,p\right)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). … …
##### 3: 19.25 Relations to Other Functions
19.25.16 $\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}\omega^{2}R_{J}\left(c-1,c-k^{2% },c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2% })}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),(\alpha^{2}-c)(c-\omega^{2})% \right),$ $\omega^{2}=k^{2}/\alpha^{2}$.
##### 4: Bille C. Carlson
The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …
##### 5: Errata
• Section 24.1

The text “greatest common divisor of $m,n$” was replaced with “greatest common divisor of $k,m$”.

• Chapter 18 Orthogonal Polynomials

The reference Ismail (2005) has been replaced throughout by the further corrected paperback version Ismail (2009).

• Figure 4.3.1

This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.

Reported 2015-11-12 by James W. Pitman.

• Section 27.20

The entire Section was replaced.

• Subsection 21.10(i)

The entire original content of this subsection has been replaced by a reference.

• ##### 6: 18.7 Interrelations and Limit Relations
###### §18.7(i) Linear Transformations
18.7.17 $U_{2n}\left(x\right)=W_{n}\left(2x^{2}-1\right),$
18.7.18 $T_{2n+1}\left(x\right)=xV_{n}\left(2x^{2}-1\right).$
If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …